238. Suppose it required to find what 156 yards will cost, if 22 yards cost 17_s._ 4_d._ This quantity, reduced to pence, is 208_d._; and if 22 yards cost 208_d._, each yard costs ²⁰⁸/₂₂_d_. But 156 yards cost 156 times the price of one yard, and therefore cost 208 208 × 156 ---- × 156 pence, or --------- pence (117). 22 22 Again, if 25½ French francs be 20 shillings sterling, how many francs are in £20. 15? Since 25½ francs are 20 shillings, twice the number of francs must be twice the number of shillings; that is, 51 francs are 40 shillings, and one shilling is the fortieth part of 51 francs, or ⁵¹/₄₀ francs. But £20 15_s._ contain 415 shillings (219); and since 1 shilling is ⁵¹/₄₀ francs, 415 shillings is 51 × 415 ⁵¹/₄₀ × 415 francs, or (117) -------- francs. 40 239. Such questions as the last two belong to the most extensive rule in Commercial Arithmetic, which is called the RULE OF THREE, because in it three quantities are given, and a fourth is required to be found. From both the preceding examples the following rule may be deduced, which the same reasoning will shew to apply to all similar cases. It must be observed, that in these questions there are two quantities which are of the same sort, and a third of another sort, of which last the answer must be. Thus, in the first question there are 22 and 156 yards and 208 pence, and the thing required to be found is a number of pence. In the second question there are 20 and 415 shillings and 25½ francs, and what is to be found is a number of francs. Write the three quantities in a line, putting that one last which is the only one of its kind, and that one first which is connected with the last in the question.[49] Put the third quantity in the middle. In the first question the quantities will be placed thus: 22 yds. 156 yds. 17_s._ 4_d._ In the second question they will be placed thus: 20_s._ £20 15_s._ 25½ francs. [49] This generally comes in the same member of the sentence. In some cases the ingenuity of the student must be employed in detecting it. The reasoning of (238) is the best guide. The following may be very often applied. If it be evident that the answer must be less than the given quantity of its kind, multiply that given quantity by the less of the other two; if greater, by the greater. Thus, in the first question, 156 yards must cost more than 22; multiply, therefore, by 156.